Sharp $$L_p$$ estimates for paraproducts on general measure spaces

Archiv der Mathematik ◽

10.1007/s00013-021-01669-y ◽

2021 ◽

Author(s):

Adam Osękowski

Keyword(s):

Function Method ◽

Bellman Function ◽

General Measure ◽

Measure Spaces

AbstractThe paper contains the identification of the $$L_p$$ L p norms of paraproducts, defined on general measure spaces equipped with a dyadic-like structure. The proof exploits the Bellman function method.

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Approximation, complete continuity, and uniform measurability of Uryson operators on general measure spaces

Nonlinear Analysis ◽

10.1016/s0362-546x(97)00595-6 ◽

1998 ◽

Vol 33 (7) ◽

pp. 715-728 ◽

Cited By ~ 2

Author(s):

M. Väth

Keyword(s):

General Measure ◽

Complete Continuity ◽

Measure Spaces

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On (L1)* for General Measure Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1963-020-6 ◽

1963 ◽

Vol 6 (2) ◽

pp. 211-229 ◽

Cited By ~ 15

Author(s):

H. W. Ellis ◽

D. O. Snow

Keyword(s):

Measurable Function ◽

Measure Space ◽

Finite Measure ◽

Signed Measure ◽

General Measure ◽

Measure Spaces ◽

Arbitrary Measure ◽

Image Position ◽

Nikodym Theorem

It is well known that certain results such as the Radon-Nikodym Theorem, which are valid in totally σ -finite measure spaces, do not extend to measure spaces in which μ is not totally σ -finite. (See §2 for notation.) Given an arbitrary measure space (X, S, μ) and a signed measure ν on (X, S), then if ν ≪ μ for X, ν ≪ μ when restricted to any e ∊ Sf and the classical finite Radon-Nikodym theorem produces a measurable function ge(x), vanishing outside e, with

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A weighted weak-type bound for Haar multipliers

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210517000415 ◽

2018 ◽

Vol 148 (3) ◽

pp. 643-658

Author(s):

Adam Osȩkowski

Keyword(s):

Maximal Operator ◽

Function Method ◽

Dual Problem ◽

Weak Type ◽

Analogous Result ◽

Type Inequality ◽

Bellman Function ◽

Weak Type Inequality ◽

Image Position

We study a weighted maximal weak-type inequality for Haar multipliers that can be regarded as a dual problem of Muckenhoupt and Wheeden. More precisely, if Tε is the Haar multiplier associated with the sequence ε with values in [−1, 1], and is the r-maximal operator, then for any weight w and any function f on [0, 1) we havefor some constant Cr depending only on r. We also show that the analogous result does not hold if we replace by the dyadic maximal operator Md. The proof rests on the Bellman function method; using this technique we establish related weighted Lp estimates for p close to 1, and then deduce the main result by extrapolation arguments.

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